An enthymeme is a particular means of expressing a syllogistic argument which
has one proposition suppressed—i.e., one proposition (either a premiss or a
conclusion) is not stated. [1]

In ordinary language, nearly all syllogistic arguments are expressed as
enthymemes. The missing proposition in these arguments is left implicit for
ease of expression and is usually easily supplied by the listener. Often,
if the missing statement were explicitly stated, the argument would lose
rhetorical effectiveness and would be thought of as “stating the
obvious.”

In some cases, the missing proposition is not explicitly stated because the
inference is only probable. It the missing premiss or conclusion were to be
explicitly supplied, the argument would be seen to be formally
invalid.

The following enthymematic example is often mistakenly attributed
to Alexis de Tocqueville: [2]

”America is great because she is good.”

Implicitly, the conclusion “America is great” logically follows
only if the doubtful premiss ”All good nations are great nations”
is assumed and added to the given premiss “She (i.e. America)
is good. Thus, when the argument is explicitly reconstructed, it
becomes

All good nations are great nations.
America is a good nation
-------------------------------------------
America is a great nation.

Note that in constructing the argument as valid, we necessarily
were restricted to a false major premiss; consequently, the argument
is unsound.

Consider this second example:

“You'll do fine, just follow your heart.”

The missing premiss necessary for validity in the argument would be
“All persons who follow their heart are persons who do
fine.”

Note that the explicit statement of the missing premiss makes
the argument valid but unsound
since the supplied premiss is clearly false. Some persons who follow
their heart do not do well.)

(All persons who follow their heart are persons who do fine.)
You are a person who follows your heart.
----------------------------------------------------------------
You are a person who does fine.

In other cases, if the missing proposition were present explicitly, the
argument might lose rhetorical force.

E.g., “Mary does well because she pays attention.”

Here, the suppressed premiss necessary for validity would be
“All people paying attention are people who do well.”
(Note that it seems reasonable that some persons who pay attention
might not do well.) And so, the argument when stated explicitly
becomes:

(All persons paying attention are people who do well.)
Mary is a person paying attention.
----------------------------------------------------------
Mary is a person who does well.

Occasionally, a proposition is suppressed in an effort to conceal the
unsoundness or the invalidity of the argument.

E.g., “No cars with internal combustion engines are energy
efficient, so no American-made cars are energy efficient.” (The missing
premiss necessary for validity here is the false premiss, "All
American-made cars are cars with internal combustion engines.) The
reconstructed argument, then looks like this:

Note: Some sources define an enthymeme as an argument in which a premiss is
missing. Nevertheless, some enthymemes omit the conclusion in order to tweak a
rhetorical effect.

E.g., “Self-absorbed people don't help charities and I know you not to
be self-absorbed.” In this psychologically manipulative reasoning, the missing
conclusion would be intended to be something like "So I'm sure you will
help.”

However, no conclusion validly follows from two negative premisses. Possibly in this case, the
conclusion was left unstated both for the reason the argument is invalid and for the
supposed rhetorically persuasive effect of appealing to one's vanity in order to
obtain help. Reconstructing the full argument, we obtain the following syllogism:

No self-absorbed people are persons who help charities.
You are not a self-absorbed person. (Note this is an E statement.)
----------------------------------------------------------------------
(You are a person who helps charities.) (Note this is an A statement.)

As mentioned above, this syllogism tests out invalid because of its exclusive
premisses.

By the principle of charity, we
should attempt to supply a missing statement that makes the argument valid
unless the context of the passage explicitly prevents such an
interpretation.

To be able to supply the missing statement requires through knowledge of
the rules for syllogisms and an understanding of the intention of the
individual advancing the argument. In the beginning, it might be helpful
to check off each syllogistic rule systematically in order to deduce the
appropriate missing proposition. Later, once the rules and fallacies become
familiar while working exercises, checking each syllogistic rule will not
be necessary to find the intended missing proposition.

Normally, during evaluation, if a proposition is intentionally supplied
making the argument invalid when such a proposition was not so intended by
the individual advancing the argument, the straw man
fallacy would be committed by the evaluator.

First, let us consider some example enthymematic arguments based on statement
forms alone. To see if these elliptical argument forms are valid we must supply the
suppressed proposition in accordance with the rules for validity.

We can systematically check each rule and its related fallacy in order
to determine the structure of the statement form necessary for validity
as follows: [3]

Rule 1: The syllogism must have
exactly three terms. The argument form already has exactly three terms:
S, P, and M, so this rule is being followed.

Rule 2: The middle term must be
distributed at least once in the premisses. The middle term is
distributed in the subject of the minor premiss (as the subject of
an A proposition), so this rule was not violated.

Rule 3: If a term is undistributed
in a premiss it cannot be distributed in the conclusion.
(Otherwise, we would be reasoning from only part of a class to a
conclusion involving the entire class.) Since the minor term S
is undistributed in its premiss, the minor term S cannot
be distributed in the conclusion or else the fallacy of illicit minor
would occur.

Rule 4: At least one premiss
must be affirmative. This rule checks out OK since the minor premiss
is affirmative.

Rule 5: If a premiss is negative,
the conclusion must also be negative. Since the major premiss
of the argument is negative, the missing conclusion must be
negative or else the fallacy of Affirmative Conclusion for a
Negative Premiss would occur.

Rule 6: If both premisses are
universal the conclusion must be universal as well. Since the
major premiss of the argument form is particular, this rule does not
apply.

Thus, from our examination of the syllogistic rules, we conclude that
the conclusion must contain both the S and P terms, and
the conclusion must be negative with the minor term S,
undistributed.

“The Body of Persuasion: A Theory of the Enthymeme”
Jeffrey Walker questions how enthymemes are defined in many rhetoric and composition
studies, develops a better characterization, and analyzes two readings first
published in the journal College English.

1. “Enthymeme” is not used in contemporary logic in the Aristotelian
sense of the word. For example, Aristotle states “Now an enthymeme
(ενθυμημα) is a syllogism starting
from probabilities or signs…” where, for him, a sign is a generally
approved demonstrative proposition) (The Basic Works
of Aristotle,
ed. Richard McKeon, Analytica Priora, trans. A.J. Jenkinson
(New York: Random House: 1970) Bk. II, Ch. 26). Here is Aristotle's example of
the former type of sign (where the logical support moves from specific to
general):

“The fact that Socrates was wise and just is a sign that the wise
are just”(Ibid, Rhetorica
trans. W. Rhys Roberts, Bk. 1 Ch. 2, 1357b).

Aristotle says this argument is
refutable because it does not form a syllogism. Rather than classifying this
argument in accordance with Aristotle's concept of “enthymeme”
contemporary usage would label the argument as an example of the informal fallacy
of converse accident. Although enthymematic arguments
are discussed here in terms of traditional formal logic, notice
that these arguments can also be taken in the rhetorical sense of being probable,
as is often done so in English rhetoric and composition
studies.↩

3. These rules and fallacies presented here are from I.M. Copi and Carl Cohen,
Introduction to Logic (Pearson, 2010). Rules in various textbooks
differ according to author, but the main procedure outlined here would, of
course, work systematically to discover the missing propositions with those rules and
fallacies as well. ↩